Wikipedia:Featured article candidates/Pi/archive1


 * The following is an archived discussion of a featured article nomination. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article candidates. No further edits should be made to this page.

The article was promoted by Ian Rose 16:39, 4 June 2012.

Pi

 * Nominator(s): Noleander (talk) 02:23, 15 May 2012 (UTC)

I am nominating this article for featured article because $\pi$ is a well-known number, famed among geeks and non-geeks alike. The article is the 483rd most-visited WP article, and it is a vital article. The article reached GA status in November 2007. It recently went through two peer reviews, one here, and another on the article's Talk page – the latter review was by user Jakob scholbach who successfully pushed the math article logarithm to FA status last year. My prior featured articles are W. E. B. Du Bois and Birth control movement in the United States. I believe the π article meets all FA criteria, and I am ready and willing to make any improvements suggested by the reviewers. Noleander (talk) 02:23, 15 May 2012 (UTC)
 * I helped push this article to GA status, and while I haven't been as involved lately, I'm excited that you've opened up the FAC. If I have a couple spare minutes during the process, I'll try to help. :) Disavian (talk) 04:22, 15 May 2012 (UTC)
 * I'll be happy to lend my hand at reviewing the article for CE, and a few other things. L1ght5h0w (talk) 13:06, 15 May 2012 (UTC)


 * Drive by comments from Jim  Jimfbleak -  talk to me?  14:58, 15 May 2012 (UTC)
 * π ≃ 355/1133 ≃ 141592920. &mdash; neither of these equalities make sense.
 * Done - That was a typo introduced about an hour ago by an overzealous editor. I've corrected it. --Noleander (talk) 15:08, 15 May 2012 (UTC)
 * I doubt the assertion that Pi day is celebrated around the world, since 3/14 is US style. Elsewhere, like the UK, the 14th day of March would be 14/3
 * Done - You are correct, the sources seem to give only US celebrations in their examples. Changed to US. --Noleander (talk) 15:08, 15 May 2012 (UTC)
 * Thanks for prompt response, I may do a proper review, but a bit tied up at present  Jimfbleak -  talk to me?  15:21, 15 May 2012 (UTC)
 * It does not make sense to celebrate Pi on other days in the rest of the world. The day was introduced in the US, and it has of course been adopted in other countries, though the extent of celebrations is probably not the same. Nageh (talk) 17:34, 19 May 2012 (UTC)

Support – Thank you for addressing my concerns. Regards, RJH (talk) 01:23, 16 May 2012 (UTC) Comments – I've been through this article a few times already and it has undergone significant improvement. There are a few emotive statements that may or may not be encyclopedic; I'll leave that for others to argue. Here's a few points that caught my eye:
 * "For this reason, most mathematicians prefer definitions of π based on calculus or trigonometry that do not rely on the circle": I'm sure it's probably true, but it's such a broad assertion that it would seem difficult to verify. Is this opinion covered by the citations on the subsequent sentence?
 * Done - The sources do mention this alternative definition, but the "most mathematicians" is not exactly in the sources. I changed it to "some mathematicians".  --Noleander (talk) 16:02, 15 May 2012 (UTC)


 * "The Greek letter π represents the ratio of a circle's circumference to its diameter": This sentence stands out because it appears redundant with the first sentence of the previous section, apart from the words "Greek letter".
 * Done - Changed to: "The symbol used by mathematicians to  represent the ratio of a circle's circumference to its diameter is the  Greek letter π. " --Noleander (talk) 16:06, 15 May 2012 (UTC)


 * I think the sentence below needs work. For one, it has two contrary sub-components that begin with 'but'. For another, "estimated to be larger than the irrationality measure of other transcendental numbers" is vague. Is it saying all transcendental numbers, or is it just saying that some well-known transendental numbers are larger than others? The latter doesn't seem particularly significant.
 * "The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known, but it is estimated to be larger than the irrationality measure of other transcendental numbers such as e or ln(2), but smaller than the measure of Liouville numbers."
 * Done - Changed to "  The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln(2), but smaller than the measure of Liouville numbers." - Let me know if that does not address  your concerns. --Noleander (talk) 16:10, 15 May 2012 (UTC)


 * "An infinite series for π that converges more rapidly than the Gregory–Leibniz series is": do we know the origin of this series?
 * No, I have not been able to find a name or origin for that series. A PR reviewer asked that 2 series be compared to illustrate convergence rates; I picked this series because it is understandable to the layman, and because it is attractive.  It is a valid series, and is documented in reliable sources.   Other series for pi that have names (or well documented origins) are much more complex and would defeat the purpose of illustrating convergence rates. So, it is a balancing act: fame vs clarity.   If anyone can find a series that has a name/origin and is simple, I'd be happy to utilize it. --Noleander (talk) 16:12, 15 May 2012 (UTC)
 * This source attributes it to Nilakantha. I haven't been able to find a better source, perhaps in part because Google limits book access. Regards, RJH (talk) 19:33, 15 May 2012 (UTC)
 * Thanks for finding that web site: at least it gives a hint about the origin, even if it is not truly a reliable source. I've posted a query on the Talk page of the Math project: with luck, someone there may have more information.  I don't think it is a show-stopper if a description of the  origin is absent.  --Noleander (talk) 20:06, 15 May 2012 (UTC)
 * Aha! It's the same formula as 16.10 from here. (n - 1)n(n + 1) = n(n2 - 1) = n3 - n. Regards, RJH (talk) 22:39, 15 May 2012 (UTC)
 * Great! Thanks for finding that. I'll update the citations accordingly. --Noleander (talk) 22:52, 15 May 2012 (UTC)


 * "that the values of [what?] within a closed boundary": is lacking some sort of clarification. A bound, continuous function perhaps? It's been a few decades since I saw that stuff, so I'm a little rusty.
 * Done - Thanks for catching that. Fixed to: " including the remarkable fact that the values of a complex function within a closed boundary are entirely determined by the values on the boundary" --Noleander (talk) 16:16, 15 May 2012 (UTC)


 * First use of "divisible" should be linked, just for clarity.
 * Done --Noleander (talk) 16:16, 15 May 2012 (UTC)


 * "Recent decades have seen a surge..." is a dated statement.
 * Done - changed this picture caption to the simpler "The record for number of memorized digits of π.". --Noleander (talk) 16:19, 15 May 2012 (UTC)

Regards, RJH (talk) 15:50, 15 May 2012 (UTC)


 * Comment Yoshida Kōyū calculated 3.16 on the soroban, while Imamura 3.162. Is this worth a mention? . Regards.-- GoP T C N 15:59, 15 May 2012 (UTC)
 * Hmm. I'm not sure. None of the sources which describe the history of pi mention these mathematicians; and the source you've provided doesn't give any details (what algorithm did they use?).  That, coupled with the fact that their precision was only 1 digit (way behind other countries in the 17th century) may make this not suitable for inclusion.  But, if others want it included, I have no objection. --Noleander (talk) 16:32, 15 May 2012 (UTC)
 * I can't do a full review, but including the information "square root of 10 had long been a common value for pi in India, China and Arabia" would be a good idea. Non-western approaches to the problem are relevant, I think. Buttonwillowite (talk) 18:10, 15 May 2012 (UTC)
 * Oops. I see that this information is included under the "Antiquity" section. Nevermind! Buttonwillowite (talk) 18:13, 15 May 2012 (UTC)


 * Question What does "The digits in the decimal representation of π appear to be random" mean? The digits of pi are not random. They can be predicted, by many of the formulae presented later on. This needs clarification.
 * The Properties section explains that in detail. It says "The digits of π appear to be random, with no observable pattern. A mathematical test for randomness is normality, meaning that all possible sequences of digits (of any given length) are equally likely."   Is that sufficient? or do you think more detail is needed?   We could remove the term "random" and use "normal" exclusively, but that would be confusing to lay readers ... the word "random" has a few meanings, and one of them is that the digits normal.   --Noleander (talk) 20:03, 15 May 2012 (UTC)
 * Ok, good. I would nevertheless prefer precision here. Simply "The digits of pi have no observable pattern" seems the best to me.
 * Now that I think of it I'm a bit concerned as well. Randomness is used in a pretty informal way here, and especially combining the words "mathematical" and "randomness" doesn't seem right. Neither does normality imply randomness, nor is normality a test to measure (statistical) randomness. Nageh (talk) 17:57, 19 May 2012 (UTC)
 * See comment below under random/normal. --Noleander (talk) 13:34, 20 May 2012 (UTC)


 * Support: Prior to the nomination of this article, I have done a FA-like review (see Talk:Pi), which did bring up a number of issues, but all of these are now fixed. So, I'm happy to support this Fa nomination. As far as I can tell, the article is factually correct and covers everything that is relevant. Jakob.scholbach (talk) 08:56, 16 May 2012 (UTC)

Comments
 * "π is an irrational number, which means that it cannot be expressed as a ratio of two integers (such as 22/7)" IMO, it needs to be clarified here that "however" pi is generally approximated as 22/7 since 22/7 is one of the most famous approximations.
 * Done - I changed it to the following: "π is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7 (where 22/7 was a commonly used approximation to π)." but that doesn't seem optimal to me.    Can you suggest a better wording? --Noleander (talk) 13:30, 19 May 2012 (UTC)
 * "...such as 22/7 and other fractions that were commonly used to approximate π." Maybe not perfect, but probably better. Nageh (talk) 18:03, 19 May 2012 (UTC)
 * Done - In lead and body. --Noleander (talk) 13:27, 20 May 2012 (UTC)


 * " In the past century," dated statement. -- Redtigerxyz Talk 06:28, 19 May 2012 (UTC)
 * Done - Changed to ". Several people have endeavored to memorize  ..." --Noleander (talk) 13:30, 19 May 2012 (UTC)

Comments
 * "mathematicians have worked strenuously to compute π to thousands and millions of digits" -> Many of the people involved in those computations are not mathematicians.
 * Done - Good catch. I tried "scientists" but a couple of them were just amateurs, so I ended up with "Despite this, people have worked strenuously to compute ...". --Noleander (talk) 13:36, 19 May 2012 (UTC)


 * "providing raw data to evaluate the randomness or normality of the digits of π" -> This suggests that "randomness" and "normality" are two different concepts, contradicting your remarks above.
 * Done - Changed to "to evaluate the randomness of the digits... ". -    "Normality" is the official term of the primary math test for randomness.  A random number generator's output can be tested for normality.  Can the digits of π be used as a random number generator?  Yes, because they appear to be normal.  Let me know if it is satisfactory now. --Noleander (talk) 13:36, 19 May 2012 (UTC)
 * Huh? You obviously have a different definition for what constitutes a random number generator. In any definition that I'm aware of unpredictability is a key requirement. Uniformity/normality is a necessary but not sufficient criterion. Nageh (talk) 18:07, 19 May 2012 (UTC)
 * I have no personal opinion on the matter: I'm just trying to capture what the major secondary sources that discuss pi for the layman say.  Those sources do use the word "random" for pi very explicitly: as in "the digits of pi appear to be random" or " ... meet statistical tests for randomness".  And the sources do say that normality is a major (the major?) test for randomness.  And  the sources say normality is generally considered to be a key attribute of randomness, but that there are some normal sequences that are not random (so n. is necessary but not sufficient for r.).  And they say that "random" sometimes means unpredictable, but also means "predictable, but digits are scattered in a hapazard way" (my paraphrase).  All that said, I'm happy to reword the sentence in question  provided that the new wording  (1)  is consistent with the sources; and (2) is understandable to lay readers.  Can someone suggest a better wording?  --Noleander (talk) 13:34, 20 May 2012 (UTC)
 * How about introducing the notion of Pseudorandomness? The sources on pi don't really get into that much, but if it helps resolve this concern, it may be a good path to take.  The text could be something like:   "Pi's digits pass have no apparent pattern and pass tests for randomness such as normality.  The digits of pi can be used as a Pseudorandom number generator since they generate a sequence of digits which meet tests for randomness, but the digits are not truly random because they are predictable."  Is that better? --Noleander (talk) 12:54, 21 May 2012 (UTC)
 * Referring to pseudo-randomness seems no less hairy, and the sentence could be quite misleading. There are basically two definition for pseudo-randomness: one (more informal) is that it satisfies certain statistical randomness tests; the other formal one from complexity theory states that it must be computationally (polynomially) indistinguishable from random data, which includes normality and unpredictability. I would rather go only with the first sentence, and replace "randomness" by "statistical randomness". Nageh (talk) 13:45, 21 May 2012 (UTC)
 * Okay, I changed the text in the Properties section to read "The digits of π have no apparent pattern and pass tests for statistical randomness such as normality;   normal means that all possible sequences of digits (of any given length) are equally likely.".  That takes the above-discussed sentence, and merges it with a pre-existing sentence which gives a brief idea what "normal" means.  Let me know if you think more needs to be done. --Noleander (talk) 13:52, 21 May 2012 (UTC)
 * I think the wording could be improved. How about this: "The digits of π have no apparent pattern and pass tests for statistical randomness such as normality; a number is called normal when all possible sequences of digits (of any given length) are equally likely." Nageh (talk) 14:05, 21 May 2012 (UTC)
 * Done --Noleander (talk) 14:10, 21 May 2012 (UTC)
 * Thanks. Nageh (talk) 14:13, 21 May 2012 (UTC)


 * "Computing a large number of digits of π does have some practical benefits" -> Simplify wording, "does have some" can simply be "has"
 * Done -- Noleander (talk) 13:36, 19 May 2012 (UTC)


 * PSLQ needs to be explained or linked
 * Done - The link for PSLQ is already there, in integer relation algorithm.  But I did add a footnote giving the origin of PSLQ as follows: "... using the PSLQ integer relation algorithm[9] found several ..." where footnote #9 says: "PSLQ means Partial Sum of Least Squares."  Normally, acronyms should be spelled out on first occurrence, but PSLQ is more of a proper name than an acronym (that is, sources that use PSLQ do not spell it out). --Noleander (talk) 13:47, 19 May 2012 (UTC)


 * are certain rational numbers that Plouffe computed -> Why is "rational numbers" suddenly linked here?
 * Done - Removed that link and linked 1st occurrence. --Noleander (talk) 13:47, 19 May 2012 (UTC)


 * the concept of a "pi hunter" is used at least 4 times. Needs explanation.
 * Done - Yes, that is too colloquial, may be hard on non-English readers. I replaced that term in all four places with plainer words. --Noleander (talk) 13:54, 19 May 2012 (UTC)


 * the "History" section has a misleading title. It's rather more "History of the computation of pi".
 * I was trying to follow the guidance of MOS:HEADINGS which says "Headings should not refer redundantly to the subject of the article, or to higher-level headings, unless doing so is shorter or clearer.". Plus, the History section includes subsection on "Irrationality and transcendence", so I'm not certain the proposed alternative is better.  If other editors endorse the longer title, I have no objection to changing it. --Noleander (talk) 13:19, 20 May 2012 (UTC)


 * " algebraic operations (powers, roots, sums, etc.)" -> I would link "algebraic operation" and clarify the "etc." There aren't that many algebraic operations.
 * Done - Eliminated the etc & included "Product" so it now reads " (powers, sums, and products)" based on the interpretation that sums includes subtraction & products  includes division.   Re the "operation": Link was done. --Noleander (talk) 13:19, 20 May 2012 (UTC)

[preceding unsigned comments were from user Randomblue]
 * "The transcendence of π means that it is impossible to solve the ancient challenge of squaring the circle" -> Are the two statements "transcendence of pi" and "cannot square the circle" equivalent? Certainly "The transcendence of pi implies that..." is true.
 * Done - Changed as suggested. --Noleander (talk) 13:23, 20 May 2012 (UTC)


 * In the list of formulae involving pi, I think the volume of an n-sphere would be very appropriate. (And would subsume many of the already given examples.)
 * I've suggested that before, but in retrospect it's likely one of those things that are primarily of interest to mathematicians because of the abstract nature of higher dimensional spheres. It's probably better just see the two and three dimensional cases covered, as they are now. The topic is covered well on the n-sphere article, so it would make sense to add that to the "See also" list. Thanks. Regards, RJH (talk) 22:59, 19 May 2012 (UTC)
 * I don't recall any secondary source on pi mentioning n-spheres.  A better place for that info might be the sub-article List of formulae involving π. --Noleander (talk) 13:23, 20 May 2012 (UTC)

Support Since my comments above, I've had another read through and I have no further concerns. I'm not a mathematician, and I felt this was pitched at just the right level for any reasonably numerate reader (and the innumerate would run off into the bushes when they saw the title &mdash; unless they were illiterate too and thought it was about food).  Jimfbleak -  talk to me?  11:01, 20 May 2012 (UTC)

Image review
 * File:Leonhard_Euler.jpg is tagged as lacking source information, and the licensing tags given are incorrect
 * Thanks for doing an image review. I fixed the license tag (changed it to Public Domain based on life-of-author +100 years).  The source is more problematic:  I was able to add information about the location of the actual painting itself, but I am unable to find out who/how it was digitized.  But, since the painter died in 1781, it is clear that any digitization should be acceptable for use in WP.  --Noleander (talk) 18:11, 21 May 2012 (UTC)


 * File:JohnvonNeumann-LosAlamos.gif: source link returns error
 * Sorry, but a dead link cannot be an argument for the license information being invalid. We also do not delete text within articles when their sources become dead. In any case, there is always the web archive. Nageh (talk) 17:39, 21 May 2012 (UTC)
 * As Nageh points out, the image is no longer located in its original location in the Los Alamos web site (I also checked and could not find a new location within that site: so it may no longer be available there at all). However, the WayBack machine does show that that was, indeed, the original source of the image. --Noleander (talk) 18:15, 21 May 2012 (UTC)
 * The picture in the article, File:JohnvonNeumann-LosAlamos.gif, is exceptional. It is an uncropped version of File:JohnvonNeumann-LosAlamos.jpg (names are the same; extensions are different). However, the JPEG version has a specific requirement for attribution of Los Alamos National Laboratories information:
 * Unless otherwise indicated, this information has been authored by an employee or employees of the University of California, operator of the Los Alamos National Laboratory under Contract No. W-7405-ENG-36 with the U.S. Department of Energy. The U.S. Government has rights to use, reproduce, and distribute this information. The public may copy and use this information without charge, provided that this Notice and any statement of authorship are reproduced on all copies. Neither the Government nor the University makes any warranty, express or implied, or assumes any liability or responsibility for the use of this information.
 * The GIF version does not have attribution.
 * Glrx (talk) 18:48, 21 May 2012 (UTC)
 * Another image from the same session at the Smithsonian: http://airandspace.si.edu/imagedetail.cfm?imageID=2689 stating "No known copyright restrictions on this image". Glrx (talk) 18:54, 21 May 2012 (UTC)
 * Glrx: thanks for that information. The GIF image is superior to the JPG image, so Ive left the GIF image in the article, and updated its Commons licensing info to incude the "los alamos PD" blurbs (copied from the JPG image).  --Noleander (talk) 18:59, 21 May 2012 (UTC)


 * File:Ramanujan.jpg: no source provided; based on information given, licensing tag may or may not be correct.
 * I was not able to justify a copyright exemption for that image, so I changed the article to use another image File:Srinivasa Ramanujan - OPC - 1.jpg which has a clearly permitted usage in WP. --Noleander (talk) 18:11, 21 May 2012 (UTC)
 * Similar images
 * http://www.math.rochester.edu/u/faculty/doug/UGpages/ramanujan.html (image on postage stamp)
 * http://www.myspace.com/sramanujan/photos/859081 (stamp apparently from 1962)
 * http://www.white-rabbit.jp/Column/essay29.html (same image, different stamp)
 * http://www.usna.edu/Users/math/meh/ramanujan.html (same/simliar to stamp image)
 * http://www.nndb.com/people/578/000240855/ (folded collar / collar tag) (possibly the source for reworked stamp image)
 * http://www.visualphotos.com/image/1x6063087/indian_mathematician_srinivasa_ramanujan (folded collar / collar tag / managed rights)
 * Glrx (talk) 19:24, 21 May 2012 (UTC)
 * It seems that the original image was scanned from an 1962 Indian postal stamp. If this is the case, the copyright on the stamp image should expire no later than by the end of this year according to commons:Commons:Stamps/Public_domain. Nageh (talk) 19:30, 21 May 2012 (UTC)
 * The 1962 stamp commemorates his 75th birthday, so the issue date was probably 22 December 1962. Glrx (talk) 19:53, 21 May 2012 (UTC)
 * http://www.indianpostagestamps.com/gallery/1962.html says issued Dec 22, 1962. Glrx (talk) 19:56, 21 May 2012 (UTC)
 * I can't count anyway. It's gonna expire by 2022, not 2012. :P Nageh (talk) 20:00, 21 May 2012 (UTC)
 * Sigh. Nominated for deletion on Commons. Glrx (talk) 20:24, 21 May 2012 (UTC)
 * To closing delegate: The article no longer uses this suspect image being discussed here.  The article was changed to use another image which has no problems. --Noleander (talk) 20:35, 21 May 2012 (UTC)

Nikkimaria (talk) 17:21, 21 May 2012 (UTC)


 * Upon further research it appears that the photography used in the stamp is taken from Ramanujan's passport. Now the question is how could we get ahold of a digital copy of that? Nageh (talk) 21:12, 21 May 2012 (UTC)
 * See . Nageh (talk) 21:23, 21 May 2012 (UTC)
 * BTW: I think the stamp/passport pic is superior, so if we can obtain a WP-safe version of it, that would be wonderful. --Noleander (talk) 21:27, 21 May 2012 (UTC)
 * http://www.imsc.res.in/~rao/ramanujan/newnow/garnlund.htm shows poor quality passport image. Image from R's Notebook (1985) may be derivative work, so it may not be free. Glrx (talk) 22:33, 21 May 2012 (UTC)
 * I agree. This book seems like the most comprehensive source on images. It appears that the wife of Subrahmanyan Chandrasekhar is the current owner of the passport photo. Nageh (talk) 00:26, 22 May 2012 (UTC)
 * Well, that is some good news. We cannot use any image that was enhanced after 1927 (e.g., head slant removed, collar altered, image effects added). Your source states that the passport photo was taken in 1919 (see page 4). The passport picture was photographically copied in India around October 1936 (page 4, 17 years after the passport was issued), but it would be a straight reproduction of a 2D image and not a derivative work. The copy of the passport picture was then printed in Hardy's Ramanujan: Twelve Lectures, 1940. Google books only gives a snippet view of Hardy. However, your source states the image on page xix is the passport photo (meaning the copy of the passport photo, and it shows the passport embossing). Ordinarily, that picture would not be a derivative work, but there may be a small argument about the shadows cast by the embossing (it is not truly a 2D image), but if the embossing is cropped out, then the image date of 1919 should apply. Well, save for the vertical striations, that I'm not sure how to interpret. At least that is how my feeble understanding of copyright goes. Therefore, I think we can copy the image out of source, crop the embossing, remove the striations, and have a free image (that we can then improve on our own). I'd like to hear comments from those more up on copyright law than I. Glrx (talk) 02:37, 22 May 2012 (UTC)
 * Maybe it is even more complicated; clock starts after the death of the author -- not the subject (or after it was made). If the author lived to, say, 1950, then the passport photo may not be in the public domain. We need an exception for government documents/passports. Glrx (talk) 03:19, 24 May 2012 (UTC)

Leonxlin's concerns


 * "The continued fraction can be used to generate the best possible rational approximation (that is, no other approximation with a smaller denominator will be closer to π)." Well, the parenthetical note is good, but the sentence should be true if the parenthetical comment were removed, which it is not, since there is no "best possible rational approximation".
 * Done - Changed sentence so the comment is now an integral part of the sentence. --Noleander (talk) 01:42, 24 May 2012 (UTC)
 * I hate to be so nitpicky, but the revised sentence still contains the independent clause "The continued fraction can be used to generate the best possible rational approximation", which is a mathematically inaccurate statement, since there is no such thing as "the best possible rational approximation". It would be like saying, "Jupiter is the closest planet to the sun; no other gas giant is as close to the sun as Jupiter." Now the situation is slightly complicated by the fact that, as I have just learned from here that best rational approximation is an accepted way of talking about rational approximations that are the best among all rational numbers with equal or less denominators. But the article the still seems quite off. The fractions 22/7, 355/113 etc. are each a best rational approximation to π. Furthermore, it seems to me that the phrase "best rational approximation" should not be used without somehow indicating that it is a technical term that is more than the sum of its parts. May I suggest the following phrasing?
 * Truncating the continued fraction at any point generates a fraction that provides an approximation for π; two such fractions (22/7 and 355/113) have been used historically to approximate the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator.
 * Leonxlin (talk) 01:05, 28 May 2012 (UTC)
 * Done - Used the suggested wording. Nitpicky is good ... this is FAC after all :-)  --Noleander (talk) 02:12, 28 May 2012 (UTC)


 * There seem to be some missing equals signs in the "Antiquity" subsection.
 * I cannot see anything missing. Could you be more specific so I can fix the problem? --Noleander (talk) 01:42, 24 May 2012 (UTC)
 * Could it be the approximately-equal signs:? Can you see the following symbol ≃ ?  If not, your platfor's font is not showing that particular character, and I can use math markup instead. --Noleander (talk) 02:36, 24 May 2012 (UTC)
 * I replaced the symbol ≃ (which is not visible on this pedestrian system) with ≈ when inexact and = when exact. Glrx (talk) 02:49, 24 May 2012 (UTC)
 * Yay! Leonxlin (talk) 01:23, 28 May 2012 (UTC)


 * "Thirty-nine digits are sufficient to support most cosmological calculations, because that is the accuracy which is necessary to calculate the diameter of the observable universe with a precision of one atom." This sentence has a cite. I do not have access to it, but I am suspicious that only the part of the sentence after "because" is confirmed in the cite, and the "sufficient to support most cosmological calculations" is someone's interpretation. The sentence may be true, but it's not clear at all that just because 39 digits suffice to calculate the diameter of the observable universe, they suffice for all other calculations. Also, I find the use of the word "support" a little strange.
 * Done - Added an attribution to the source, to make it clear there is some interpretation happening. There are two sentences that use that source (Arndt, p 17):   "According to Jörg Arndt, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy which is necessary to calculate the diameter of the observable universe with a precision of one atom. Accounting for additional digits needed to compensate for computational round-off errors, Arndt concludes that a few hundred digits would suffice for any scientific application."  - That is a direct paraphrase (but not too close :-) of what Arndt says.  Also, I changed "support" to "perform". --Noleander (talk) 01:42, 24 May 2012 (UTC)
 * Again, I hate to be difficult. But I would be much happier if we got rid of the second sentence. I don't think Arndt's speculation merits a whole sentence here. (A quick Google search, and I have found nothing to lend him any repute in math or physics. Then again, he is published and I am not.) Also, what about Haenel? Leonxlin (talk) 01:23, 28 May 2012 (UTC)
 * Done - Removed that second sentence from the article body; (kept it in a footnote so curious readers can still enrich themselves).  Also:  added Haenel so it now reads: "According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient  ..." --Noleander (talk) 02:18, 28 May 2012 (UTC)


 * The first two paragraphs of the section on spigot algorithms are a bit awkward. The definition of a spigot algorithm is repeated at least twice, and the way "characterized" is used is strange.
 * Done - Removed one of the defnitions. Changed "characterized" to "The algorithms are called ...". --Noleander (talk) 01:45, 24 May 2012 (UTC)
 * Yay! Leonxlin (talk) 01:23, 28 May 2012 (UTC)


 * I am concerned about the science sections, and feel that an expert should look over it to make sure that it's coherent. I'm no physicist, but I would have expected Coulomb to be mentioned before Heisenberg, since that's the order a student would probably see them. (Or is there some other reason why the formulae there are ordered the way they are?) And why doesn't, say, Biot-Savart show up? Is there a good reason the Fourier transform appears under "Engineering and geology" rather than "Physics"? I'm not qualified to judge whether this list is not just a bunch of random formulae, but I have some suspicions.
 * The π article is a top-level article which summarizes about ten child sub-articles. The Usage section you are referring to is summarizing the List of formulae involving π sub-article which is, indeed, a list of formulae.   This list was pretty well reviewed during the Peer Review process, and some items were removed since (1) the formulae are not directly relevant to the importance or history of pi; and (2) it is impossible to come up with a good litmus test for which should be included or excluded.   There are two extremes: omitting the list entirely; or  including all of  List of formulae involving π .  Niether extreme is proper, and the article is striving for a good middle ground.  Regarding Biot Savart:  that used to be in the article, but a Peer Reviewer explicitly asked that it be removed from the article, so I'm a bit reluctant to restore it, but I can if you really think it is important.  As for the sequence: the sources are very skimpy in this area, and any choherent ordering is feasible. Regarding Fourier transform: there are several items that could appear in either of two subsections, in those cases, I chose the subsection that seemed most appropriate.   If you think  Fourier transform should be moved to Physics, that can be easily done. --Noleander (talk) 01:53, 24 May 2012 (UTC)
 * Strongly object moving FT to the physics section. ;) FT is fundamental to signal processing, which is of primary importance to communication technologies and computer science applications. Nageh (talk) 09:18, 24 May 2012 (UTC)


 * "π is a transcendental number, which means that it is not the solution of any non-constant polynomial with rational coefficients, such as: $$ \frac{x^5}{120} - \frac{x^3}{6} + x = 0.$$" This polynomial, which is an example of something π is not a solution to (the only root is 0), and which does not appear to be special in any way, really should not be given so much space. (In the article, it gets its own line.)
 * Done - I made the formula smaller and put it in-line. Can you check and see if it is okay, or is too small?  WP has a limitation that there are only two sizes available for formulae: in my opinion, one is too large, and the other too small :-)   So this is a recurring problem that is not specific to this article.  That particular formula was specifically added during the Peer Review process at the request of the reviewer, so I am reluctant to remove it entirely. --Noleander (talk) 01:57, 24 May 2012 (UTC)


 * The "ε" used in the paragraph about the Mandelbrot set should be "e", I think.
 * The source (Aaron Klebanoﬀ) uses "ε"  so I followed that  in the article.  If you can provide some rationale for "e", we can consider going that route. --Noleander (talk) 02:01, 24 May 2012 (UTC)
 * Well my concern is just that I have no idea what "ε" is supposed to mean. If it is the base of the natural logarithm, then I would recommend "e" or "$e$". Leonxlin (talk) 18:22, 31 May 2012 (UTC)
 * ε is a small number that tends to zero. I've rewritten it to clarify this and added a link for the source.-- JohnBlackburne wordsdeeds 20:29, 31 May 2012 (UTC)


 * Some dates (e.g., "100 BC") need non-breaking spaces.
 * Done - put nbsp before all "BC"s. --Noleander (talk) 02:01, 24 May 2012 (UTC)

Leonxlin (talk) 21:55, 23 May 2012 (UTC)

Leonxlin (talk) 19:00, 31 May 2012 (UTC) Source spotchecks
 * "This formula was a breakthrough for people who calculated π because ..." An odd phrase.
 * Done – I changed it to "a breakthrough in calculating π". —David Eppstein (talk) 21:33, 31 May 2012 (UTC)

The following footnote numbers refer to those in this version of the article.

Leonxlin has started checking multiples of 5. The ones he skipped are the ones he doesn't have access to. Leonxlin (talk) 20:57, 23 May 2012 (UTC)


 * #5 Arndt & Haenel 2006, p. 165.: The paragraph citing this footnote (as well as #6 and #7) says that "The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones", later mentioning "speculation that Machin may have employed the Greek letter before Jones". Arndt and Haenel, however, seem to give more weight to this speculation: they say that "The person who invented the symbol π is assumed to have been the Englishman William Jones", but then later, "The symbol π must therefore be attributed to the same man who had already earned his place in history with his arctan formula [Machin]" (link to Google Books in the article). Wolfram doesn't mention Machin at all on this issue, though. Leonxlin (talk) 20:57, 23 May 2012 (UTC)
 * Only one source (Arndt) mentions the possibility that Machin used the letter before Jones, and they suggest it rather strongly. But no other sources echo that.  So I tried to capture that by characterizing Arndt's thesis using the wording "Jones writes that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones."   Is this adequate?  Or can you suggest a better wording?  Do you think Arndt should be mentioned in the article here? --Noleander (talk) 02:06, 24 May 2012 (UTC)
 * This makes sense. Leonxlin (talk) 19:00, 31 May 2012 (UTC)


 * #10 Mayer. Check. Leonxlin (talk) 20:57, 23 May 2012 (UTC)


 * #15 Arndt & Haenel 2006, p. 21 BAD BAD BAD: The article reads "The digits of π have no apparent pattern and pass tests for statistical randomness such as normality". Arndt and Haenel say this: "No one has succeeded in proving either than π is normal or that it is not". Leonxlin (talk) 20:57, 23 May 2012 (UTC)
 * There are three footnotes that apply: #15, #16, and #17.  It looks like a sentence got split and the correct footnote (#17) was separated from the accompanying text.  I'll fix that. --Noleander (talk) 22:28, 23 May 2012 (UTC)
 * I think you are confusing statistical tests with the actual normality of a number. The sentence can easily be referenced to page 23 of the book. However, I would suggest saying "...measuring normality" rather than "...such as normality"; normality is not a randomness test but a property. Nageh (talk) 22:29, 23 May 2012 (UTC)
 * Done - Aligned the footnotes with the sentences (and included page 23 in addition to 21 and 22). Also improved wording per Nageh suggestion to "The digits of π have no apparent pattern and pass tests for statistical randomness including tests measuring normality" (because some of the statistical tests were not related to normality). --Noleander (talk) 02:13, 24 May 2012 (UTC)
 * I now understand where my confusion comes from. The definition of normality does not imply that all sequences are equally "likely", meaning, that their probabilities are equal and independent, as the article claimed. Instead, the definition only states that the frequency of all sequences appear equally often asymptotically. This is also wrong in the normal number article. I have reworded it in the pi article to: "a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often." Nageh (talk) 08:55, 24 May 2012 (UTC)
 * Never mind, I was thinking too complicated. Though, do you think the new wording is more clear than the previous one? If not, please revert my edit in the article. Nageh (talk) 09:12, 24 May 2012 (UTC)
 * The new wording is fine. --Noleander (talk) 13:15, 24 May 2012 (UTC)
 * (Please forgive the tone of the subsequent sentences and correct any misunderstandings I have; I'm just trying to get as much done as possible quickly. I am not an expert on this subject.) It seems weird to me to say that π "passes ... tests measuring normality". If normality is a property of a number, a yes or no, than what does it mean to "measure" normality? And where on page 23 of Arndt and Haenel does it say that any tests have been done on π regarding normality? The word "normal(ity)" does not even appear on that page. To say a number is normal seems like a very strong statement about all of its digits, so that if statistical tests are done on the first X digits of π, no matter how large X is, that will say exactly nothing about the normality of π. Why not simply get rid of the "including normality" part of that sentence? Leonxlin (talk) 19:00, 31 May 2012 (UTC)
 * It is a yes-or-no property, so maybe "measuring" is the wrong word and we should simply say "tests for normality". However, note that the answer is not known, so these statistical tests only provide indication to a particular answer. I don't understand your issue with the references: Did you read the last paragraph of page 22 of Arndt's book? That provides the connection to the statistical tests discussed at page 23. Nageh (talk) 20:01, 31 May 2012 (UTC)
 * Changed wording to: "The digits of π have no apparent pattern and pass tests for statistical randomness including tests for normality". --Noleander (talk) 13:54, 1 June 2012 (UTC)


 * #30 Arndt & Haenel 2006, pp. 168–169: Check. Leonxlin (talk) 20:57, 23 May 2012 (UTC)


 * #50 Roy 1990, pp. 101–102 and Arndt & Haenel 2006, pp. 185–186. Check. (Though I'm not sure about the Roy page numbers; I don't have the same version of the article.) Leonxlin (talk) 20:57, 23 May 2012 (UTC)


 * #90 NumberWorld.org. Check. Leonxlin (talk) 20:57, 23 May 2012 (UTC)


 * #100 Bellard, Fabrice. Wikipedia currently says "Between 1998 and 2000, the distributed computing project PiHex used Bellard's formula (a modification of the BBP algorithm) to compute the quadrillionth (1015th) bit of π, which turned out to be 0." In the cited article, Bellard says nothing about PiHex or the quadrillionth bit of π. Leonxlin (talk) 20:57, 23 May 2012 (UTC)
 * Done - Thanks for finding that ... there were two sources, and one of them (Arndt, p 20) got dropped from the footnote.  I've added the missing source. --Noleander (talk) 02:20, 24 May 2012 (UTC)


 * #105, #110. Don't have the sources, but I can personally verify that the claims are true. Leonxlin (talk) 20:57, 23 May 2012 (UTC)

Per the above concerns, unfortunately I will have to Oppose. Leonxlin (talk) 21:55, 23 May 2012 (UTC)
 * Thanks for the review. It looks like all the issues you found are resolvable.  I'll start working on them promptly. --Noleander (talk) 22:24, 23 May 2012 (UTC)
 * Leonxlin: I've addressed most of the issues you raised, and posed a couple of questions to get clarification.  --Noleander (talk) 02:27, 24 May 2012 (UTC)
 * Opposed rescinded, due to concerns being largely addressed. Leonxlin (talk) 00:09, 28 May 2012 (UTC)

'''Lead is saucy, spunky, and wonderful. It makes me quiver with nerd joy. Cryptic C62 · Talk''' 17:48, 31 May 2012 (UTC)


 * Lead should include pronunciation
 * Done - Although I only put the IPA in the lead; the body has both IPA and respelling.


 * "The number π is a mathematical constant that is the ratio of a circle's circumference to its diameter." Shouldn't "a circle" be "any circle"?
 * "a circle" sounds better to my ear. Comparing to other articles: the 1st sentence in perimeter is "A perimeter is a path that surrounds an area."   If you can cite  a few well-written articles that use "any" I could perhaps be persuaded. --Noleander (talk) 23:55, 31 May 2012 (UTC)
 * Hmm, on second thought, "a circle" doesn't seem that bad. I'm not sure what bugged me about it the first time. Take note, however, that comparing an issue to one in another article, even another FA, is not sufficient justification for allowing the issue to persist. All it does is highlight a way in which the other article can be improved. This diff gives more details, if you're curious. --Cryptic C62 · Talk 13:24, 1 June 2012 (UTC)
 * Agree with you point; I was just trying to say that the MOS was silent on that particular issue, and I could not find any examples that were consistent your proposal (didn't mean to suggest that examples would be determinative). --Noleander (talk) 13:48, 1 June 2012 (UTC)


 * Inconsistency:
 * Lead: "Scientific applications require no more than a few hundred digits of π"
 * Body: "For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations"
 * Done - Changed lead too " about 40 digits". Let me know if you can think of a better wording. --Noleander (talk) 23:55, 31 May 2012 (UTC)
 * I changed it to "Scientific applications generally require no more than 40 digits". You dig? --Cryptic C62 · Talk 13:24, 1 June 2012 (UTC)
 * Yes, that is even better. --Noleander (talk) 13:48, 1 June 2012 (UTC)


 * "The peculiar properties of π" Does this refer to the properties described in the first paragraph? I don't see how any of these are peculiar. There are, after all, infinitely many transcendental numbers.
 * Done - That is a good point, changed to "The widespread use of π in science and engineering has ..." --Noleander (talk) 23:55, 31 May 2012 (UTC)


 * I am of the opinion that the lead suffers from WP:recentism. It places too much emphasis on modern pop-culture shenaniganry and not enough emphasis on the early history. The only information presented about the early history is a shopping list of mathematicians who were interested in the number, whereas world records are mentioned twice.
 * Done - Added some more historical detail to the lead. I kept the names of important mathematicians in the lead, but interspersed in the historical sentences.  --Noleander (talk) 00:18, 1 June 2012 (UTC)
 * Much better! Thanks a bundle for doing this. And, more generally, thanks for taking the time to built this article. I imagine it was not easy. --Cryptic C62 · Talk 13:24, 1 June 2012 (UTC)
 * Thanks for the review. --Noleander (talk) 13:48, 1 June 2012 (UTC)

Delegate notes
 * Generally a paragraph in the body of the article should end with a citation, unless summarising cited info to follow -- you seem to maintain that standard everywhere except in the first para of Name.
 * The sentence you are referring to is "The lower-case letter π (or π in sans-serif font) is not to be confused with the capital letter Π, which denotes a product of a sequence." I don't have a source for that.  It is comparable to a disambiguation hatnote at the top of the article: an aside to the reader which sheds some light on other topics with similar names.  I can certainly remove that sentence, but I wonder if this is one of those situations where an exception can be made? --Noleander (talk) 16:22, 4 June 2012 (UTC)
 * No, that's fair enough. Cheers, Ian Rose (talk) 16:32, 4 June 2012 (UTC)


 * On the other hand, I don't see the need to repeat the same citation after each sentence, as you do in the last para of Name for instance (an exception might be where a quote has to be clearly attributed) -- although if it's your preferred style and doesn't annoy the reviewers I shan't make a fuss about it... Cheers, Ian Rose (talk) 15:47, 4 June 2012 (UTC)
 * Yeah, that is my style.  I do that for a couple of reasons: (1) in case the sentences get rearranged in the future, the cites will go with them (see WP:INTEGRITY); and (2) it is a common practice in articles covering controversial topics, so I've developed that habit.   But I would not object to an editor removing those footnotes.  --Noleander (talk) 16:22, 4 June 2012 (UTC)
 * Heh, I'm probably more immune to the rearranging issue as "my" articles are pretty specialised and don't tend to attract a lot of edits. One like this on the other hand could well benefit from the insurance of having more consecutive duplicate citations that might otherwise be warranted. So, again, fair enough. Anyway, as things have been quiet here for a few days and we have the requisite checks and support, I think we can wrap this up... Cheers, Ian Rose (talk) 16:32, 4 June 2012 (UTC)


 * The above discussion is preserved as an archive. Please do not modify it. No further edits should be made to this page.